Alan Turing Quotes

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity.

if you do not appreciate the mathematics, you cannot see, among the great variety of facts, that logic permits you to go from one to the other

Mathematics seems to endow one with something like a new sense.

Mathematics is a language plus reasoning. It's like a language plus logic. Mathematics is a tool for reasoning.

I don't really get things very... intuitively. I mean, I don't immediately understand things. The only way I really get it is by writing it down.

Even the greatest mathematicians, the ones that we would put into our mythology of great mathematicians, had to do a great deal of leg work in order to get to the solution in the end.

Logic is a wonderful thing but doesn't always beat actual thought.

Mathematical reasoning may be regarded.

Geometry is beautifully logical, and it teaches you how to think and prove that things are so, step by step by step. Proofs are excellent lessons in reasoning. Without logic and reasoning, you are dependent on jumping to conclusions or - worse - having empty opinions.

A dozen more questions occurred to me. Not to mention twenty-two possible solutions to each one, sixteen resulting hypotheses and counter-theorems, eight abstract speculations, a quadrilateral equation, two axioms, and a limerick. That's raw intelligence for you.

Being exposed to theory, stimulated by a basic love of concepts and mathematics, was a marvelous experience.

The knowledge of the theory of logic has no tendency whatever to make men good reasoners.

I don't care much for equations myself. This is partly because it is difficult for me to write them down, but mainly because I don't have an intuitive feeling for equations.

Thinking - in particular abstract thinking, which most of us are introduced to through the study of mathematics and literature - helps us learn that we can become problem solvers.

Maths should be more practical and more conceptual, but less mechanical.

An ounce of algebra is worth of a ton of verbal argument.

Solving wits and puzzles, in a way, helps to develop wit and ingenuity.

The main functions of intelligence, that of inventing solutions and that of verifying them, do not necessarily involve one another. The first partakes of imagination; the second alone is properly logical.

As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.

Beginning to reason is like stepping onto an escalator that leads upward and out of sight. Once we take the first step, the distance to be traveled is independent of our will and we cannot know in advance where we shall end.

Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature.

Science is not a substitute for common sense, but an extension of it.

Address to Albert Einstein: You are not thinking. You are merely being logical.

It's strange how few of the world's great problems are solved by people who remember their algebra.

The analytical power should not be confounded with simple ingenuity; for while the analyst is necessarily ingenious, the ingenious man is often remarkably incapable of analysis.